Group #16777120 — subgroups

Subgroups — #16777120

Order 3328.

266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131197	C₂	1	2	yes	maximal
2	C₄×#88043	C₂	1	1	yes	maximal, characteristic
3	C₄³×C₁₃	C₄	1	1	yes	characteristic
4	C₂×#88043	—	2	2	no	—
5	C₂×C₄×#1055	C₄	1	1	yes	characteristic
6	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
7	C₂²⋊#6092	—	2	2	no	—
8	#11718231	—	2	2	no	—
9	C₄⋊#6092	C₄	1	4	yes	complemented
10	C₂×C₄²×C₁₃	—	2	1	no	—
11	C₂×C₄²×C₁₃	—	2	1	no	—
12	C₂×C₄²×C₁₃	C₈	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×#1055	—	2	1	no	—
15	C₂²×#1055	D₄	1	2	yes	—
16	C₂×#6092	—	2	4	no	—
17	#88043	—	2	8	no	—
18	C₄×#1055	—	2	1	no	—
19	C₄×#1055	C₈	1	2	yes	—
20	C₄×#1055	C₂×C₄	1	2	yes	—
21	C₄×#1055	D₄	1	1	yes	characteristic
22	C₄×#1055	Q₈	1	1	yes	characteristic
23	#7149	—	13	1	no	maximal, Sylow
24	C₂²×C₄×C₁₃	—	2	2	no	—
25	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
28	C₄²×C₁₃	—	2	4	no	—
29	C₄²×C₁₃	—	4	2	no	—
30	C₄²×C₁₃	—	4	2	no	—
31	C₄²×C₁₃	C₄²	1	1	yes	characteristic
32	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
35	C₂×#1055	—	2	4	no	—
36	C₂×#1055	—	2	2	no	—
37	C₂×#1055	—	2	4	no	—
38	C₂×#1055	C₂²⋊C₄	1	2	yes	—
39	C₂×#1055	C₄⋊C₄	1	2	yes	—
40	#6092	—	4	4	no	—
41	#2163	—	13	2	no	—
42	C₄×#97	—	13	1	no	—
43	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	2	4	no	—
48	C₂×C₄×C₁₃	—	4	2	no	—
49	C₂×C₄×C₁₃	—	4	2	no	—
50	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
56	#1055	—	2	4	no	—
57	#1055	—	2	2	no	—
58	#1055	#104	1	4	yes	—
59	C₂×#97	—	26	2	no	—
60	C₂×C₄×C₈	—	13	1	no	—
61	C₂×C₄×C₈	—	13	1	no	—
62	C₄³	#261	1	1	yes	characteristic
63	C₄⋊C₁₆	—	13	4	no	—
64	C₂²⋊C₁₆	—	26	2	no	—
65	#346	—	26	2	no	—
66	C₂²×C₁₃	—	2	1	no	—
67	C₂²×C₁₃	—	2	1	no	—
68	C₂²×C₁₃	#336	1	1	yes	characteristic
69	C₂²×C₁₃	#349	1	2	yes	—
70	C₄×C₁₃	—	2	1	no	—
71	C₄×C₁₃	—	2	1	no	—
72	C₄×C₁₃	—	2	1	no	—
73	C₄×C₁₃	—	2	1	no	—
74	C₄×C₁₃	—	4	4	no	—
75	C₄×C₁₃	#339	1	2	yes	—
76	C₄×C₁₃	#347	1	2	yes	—
77	C₂²×C₈	—	13	2	no	—
78	C₂²×C₈	—	26	1	no	—
79	C₂×C₄²	—	2	1	no	—
80	C₂×C₄²	—	2	1	no	—
81	C₂×C₄²	#1057	1	2	yes	—
82	C₂×C₄²	C₂×#261	1	1	yes	characteristic
83	C₂×C₁₆	—	26	4	no	—
84	#97	—	26	8	no	—
85	C₄×C₈	—	13	2	no	—
86	C₄×C₈	—	13	2	no	—
87	C₄×C₈	—	13	1	no	—
88	C₄×C₈	—	13	1	no	—
89	C₄×C₈	—	26	1	no	—
90	C₂×C₁₃	—	2	2	no	—
91	C₂×C₁₃	#1276	1	1	yes	characteristic
92	C₂×C₁₃	#1375	1	1	yes	characteristic
93	C₂×C₁₃	#1377	1	1	yes	characteristic
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
96	C₂²×C₄	#6122	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
98	C₂×C₈	—	13	2	no	—
99	C₂×C₈	—	13	2	no	—
100	C₂×C₈	—	26	4	no	—
101	C₂×C₈	—	26	4	no	—
102	C₂×C₈	—	26	2	no	—
103	C₄²	—	2	4	no	—
104	C₄²	—	4	2	no	—
105	C₄²	—	4	2	no	—
106	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
107	C₄²	#6118	1	1	yes	characteristic
108	C₄²	C₄×#261	1	1	yes	characteristic
109	C₄²	C₄⋊#261	1	1	yes	characteristic
110	C₁₆	—	52	4	no	—
111	C₁₃	#7149	1	1	yes	characteristic, complemented, Sylow
112	C₂³	C₂²⋊#1057	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₄×#1057	1	1	yes	characteristic
120	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
121	C₂×C₄	#88110	1	1	yes	characteristic
122	C₂×C₄	#88111	1	1	yes	characteristic
123	C₂×C₄	#88112	1	1	yes	characteristic
124	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
125	C₈	—	13	4	no	—
126	C₈	—	26	4	no	—
127	C₈	—	26	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#11718284	1	2	yes	—
131	C₂²	#11718290	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	4	no	—
137	C₄	#11718232	1	2	yes	—
138	C₄	#11718234	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	#13129664	1	1	yes	characteristic
141	C₂	#13129877	1	1	yes	characteristic
142	C₂	#13129883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131197

C₂

yes

maximal

C₄×#88043

C₂

yes

maximal, characteristic

C₄³×C₁₃

C₄

yes

characteristic

C₂×#88043

—

C₂×C₄×#1055

C₄

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

C₂²⋊#6092

—

#11718231

—

C₄⋊#6092

C₄

yes

complemented

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₈

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂²×#1055

—

C₂²×#1055

D₄

yes

—

—

—

—

C₈

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

D₄

yes

characteristic

C₄×#1055

Q₈

yes

characteristic

#7149

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

M₁₆

yes

characteristic

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₄²

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

—

—

—

C₂²⋊C₄

yes

—

C₂×#1055

C₄⋊C₄

yes

—

#6092

—

#2163

—

C₄×#97

—

C₂³×C₁₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#95

yes

characteristic

C₂×C₄×C₁₃

C₄×C₈

yes

characteristic

C₂×C₄×C₁₃

#97

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

—

yes

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

#261

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#336

yes

characteristic

C₂²×C₁₃

#349

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#339

yes

—

C₄×C₁₃

#347

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#1057

yes

—

C₂×C₄²

C₂×#261

yes

characteristic

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂²⋊#261

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

D₁₃⋊C₈

yes

characteristic

107

C₄²

#6118

yes

characteristic

108

C₄²

C₄×#261

yes

characteristic

109

C₄²

C₄⋊#261

yes

characteristic

110

C₁₆

—

111

C₁₃

#7149

yes

characteristic, complemented, Sylow

112

C₂³

C₂²⋊#1057

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄×#1057

yes

characteristic

120

C₂×C₄

C₄⋊#1057

yes

characteristic

121

C₂×C₄

#88110

yes

characteristic

122

C₂×C₄

#88111

yes

characteristic

123

C₂×C₄

#88112

yes

characteristic

124

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

#11718284

yes

—

131

C₂²

#11718290

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#11718232

yes

—

138

C₄

#11718234

yes

—

139

C₂

—

140

C₂

#13129664

yes

characteristic

141

C₂

#13129877

yes

characteristic

142

C₂

#13129883

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#261)⋊C₄
commutator	C₂×C₄×C₁₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#261
Fitting	C₄³×C₁₃	C₄
socle	C₂²×C₁₃	#336
prime radical	C₂³×C₁₃	C₂²⋊C₈
cosocle	C₂×C₄×#1055	C₂²
hypercenter	C₄³	#261
hypocenter	C₁₃	#7149
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

(C₂×#261)⋊C₄

commutator

C₂×C₄×C₁₃

C₄×C₈

Frattini

C₂×C₄²

C₂×#261

Fitting

C₄³×C₁₃

C₄

socle

C₂²×C₁₃

#336

prime radical

C₂³×C₁₃

C₂²⋊C₈

cosocle

C₂×C₄×#1055

C₂²

C₄³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
9. C₄⋊#6092	C₄	semidirect
111. C₁₃	#7149	semidirect

H ≅ G/N

type

9. C₄⋊#6092

C₄

semidirect

111. C₁₃

#7149

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13131197	2
C₄×#88043	2
#7149	13

subgroup

index

#13131197

C₄×#88043

#7149

Subgroups — #16777120

Conjugacy classes of subgroups

Complements

Maximal subgroups